The extremal function for partial bipartite tilings

For a fixed bipartite graph HH and given α∈(0,1)α∈(0,1), we determine the threshold TH(α)TH(α) which guarantees that any nn-vertex graph with at least TH(α)n2 edges contains (1−o(1))αv(H)n vertex-disjoint copies of HH. In the proof, we use a variant of a technique developed by Komlós [J. Komlós, Tiling Turán theorems, Combinatorica 20 (2) (2000) 203–218].